MHF3202: Sets and Logic — Florida Course Repo

Course Description

MHF3202 Sets and Logic is the transition course in a mathematics degree — the point at which a student stops computing and starts proving.

The course is offered at approximately five Florida institutions, including Florida Atlantic University, Indian River State College, the University of Florida, the University of North Florida and the University of West Florida.

The University of West Florida titles it Set Theory and Mathematical Logic, places it in the College of Science and Engineering, Department of Mathematics and Statistics at 3 semester hours, requires MAC 2312 (with an asterisk indicating concurrency is permitted), and describes basic set theory, propositional calculus, predicate calculus, and methods of mathematical proof. It meets the College-Level Computation Skills Requirement.

This is one of the most pedagogically significant courses in an undergraduate mathematics curriculum, and its purpose is not obvious from the title. Calculus, differential equations and linear algebra are largely computational — a student learns procedures and applies them, and the answer is right or wrong. Upper-division mathematics is almost entirely proof — real analysis, abstract algebra, topology and number theory ask you to establish that something is true, in general, from stated assumptions.

That is a genuine discontinuity, and it is where a substantial number of mathematics majors discover the subject is not what they thought it was. Some find that they much prefer it; others find they were good at computation and are not interested in proof. The transition course exists to make that discovery happen deliberately and early, with support, rather than accidentally in the middle of real analysis.

The three components serve one purpose. Logic supplies the rules of inference and — more practically — the ability to negate a statement correctly, which is what indirect proofs require. Set theory supplies the vocabulary in which nearly all modern mathematics is written: functions, relations, cardinality and the language of collections. Proof technique is the skill itself. None of the three is the subject; the subject is learning to write mathematics that establishes something.

The honest difficulty is that proof cannot be taught procedurally. There is a repertoire of standard strategies, and knowing them is necessary and not sufficient — finding the proof of an unfamiliar statement requires understanding why it is true, and that is a form of mathematical maturity that develops slowly and mostly by struggling. Students who expect a method to apply are frustrated for several weeks. The frustration is the course working, and it is worth knowing that in advance.

Learning Outcomes

Required Outcomes

Use propositional logic — connectives, truth tables, tautologies and logical equivalence — fluently.
Translate mathematical statements into symbolic form and back.
Use predicate logic — quantifiers, bound and free variables, and nested quantification.
Negate quantified statements correctly, including statements with multiple quantifiers.
Distinguish a statement from its converse, inverse and contrapositive, and know which is logically equivalent to which.
Construct direct proofs.
Construct proofs by contraposition and by contradiction, and explain when each is appropriate.
Construct proofs by cases and by counterexample for disproof.
Construct proofs by mathematical induction, ordinary and strong.
Prove existence and uniqueness statements.
Prove biconditional statements and set equalities.
Use set notation and operations — union, intersection, complement, difference, power set, Cartesian product — and prove set identities.
Work with indexed families of sets and generalised unions and intersections.
Define and analyse relations, including equivalence relations, equivalence classes and partitions, and partial orders.
Define and analyse functions set-theoretically; prove that a function is injective, surjective or bijective; work with images, preimages, composition and inverses.
Explain cardinality, distinguish finite, countably infinite and uncountable sets, and reproduce the standard results.
Write mathematics clearly — with correct notation, complete sentences, stated quantifiers and explicit logical structure.
Read a proof critically and identify a gap or an error in a purported proof.

Optional Outcomes

Explain the axiomatic method and the role of axioms, definitions and theorems.
Work with the Zermelo-Fraenkel axioms and the axiom of choice in outline.
Explain Russell's paradox and the motivation for axiomatic set theory.
Prove basic results in elementary number theory as proof practice.
Apply combinatorial counting arguments.
Explain ordinals and cardinal arithmetic in outline.
Explain Gödel's incompleteness theorems at an informal level.
Use a proof assistant or typeset proofs in LaTeX.

Major Topics

Required Topics

Propositional logic. Statements and truth values; the connectives and their truth tables; the conditional and the standing student difficulty that a false antecedent makes a conditional true, which is a convention with reasons and which must be accepted before anything else works; logical equivalence and the key equivalences — De Morgan's laws, contraposition, material implication; tautology, contradiction and contingency; converse, inverse and contrapositive, and the fact that a statement is equivalent to its contrapositive but not to its converse, which is the single most consequential equivalence in the course because it licenses proof by contraposition; valid argument forms and the fallacies.
Predicate logic and quantifiers. Predicates, domains, free and bound variables; the universal and existential quantifiers; translating mathematical statements, which is where the course's first real difficulty appears — the definition of a limit or of continuity is a nested quantified statement, and a student who cannot parse it cannot use it; multiple and nested quantifiers, and the crucial fact that order matters — "for every ε there exists a δ" and "there exists a δ such that for every ε" are different claims, and the difference is exactly the difference between continuity and uniform continuity; negation of quantified statements, which is the most practically important skill in the course: to prove something by contradiction you must first state its negation correctly, and the rule that negation swaps the quantifiers and negates the body is applied constantly for the rest of a mathematics degree.
Proof techniques — the core. What a proof is and what it is for; the anatomy of a proof and the conventions of mathematical writing; direct proof — assume the hypothesis, derive the conclusion; proof by contraposition and recognising when it is easier, typically when the hypothesis is a negation; proof by contradiction, its power and the caution that it is over-used by beginners — many contradiction proofs are direct proofs with unnecessary scaffolding; proof by cases and the requirement that the cases be exhaustive; disproof by counterexample, and the asymmetry that a universal claim falls to one counterexample while an existential claim requires a construction; existence proofs, constructive and non-constructive, and why the distinction matters; uniqueness proofs and the standard "suppose two and show they are equal" move; biconditional proofs in both directions.
Mathematical induction. The principle and why it is valid; base case and inductive step, and the recurring error of assuming what is to be proved rather than assuming the statement at n and proving it at n+1; strong induction and when it is needed; induction on structures other than the integers; the well-ordering principle and its equivalence to induction; recursive definitions and proofs about them; the standard exercises — summation formulas, divisibility, inequalities — which are practice for the technique rather than results of interest in themselves.
Set theory. Sets, elements and the distinction between membership and subset, which students conflate constantly and which is a genuine source of error; set-builder notation; the empty set and its properties, including that it is a subset of everything; operations — union, intersection, complement, difference, symmetric difference; the power set and its cardinality; Cartesian products; proving set identities, both by the element-chasing method (show each side is a subset of the other) and by the algebra of sets — the element-chasing proof is the one that matters because it is where set proofs connect to logic; indexed families and generalised unions and intersections, which are the notation used throughout analysis; Russell's paradox and why unrestricted comprehension fails, which motivates axiomatic set theory.
Relations. Relations as sets of ordered pairs; the properties — reflexive, symmetric, antisymmetric, transitive; equivalence relations, and the fundamental theorem that an equivalence relation on a set partitions it into equivalence classes and that every partition arises this way — which is one of the most reused ideas in all of mathematics, appearing in modular arithmetic, quotient groups, quotient spaces and function spaces; partial and total orders, upper and lower bounds, maxima and suprema; composition and inverses of relations.
Functions, defined properly. A function as a special kind of relation, which is frequently the first rigorous definition a student has met after years of using functions informally; domain, codomain and range, and the distinction between codomain and range, which matters for surjectivity; injective, surjective and bijective, and proving each — the standard techniques are worth internalising because they recur; images and preimages of sets, and the results about how they interact with unions and intersections, which are classic proof exercises and are genuinely subtle; composition and its properties; inverse functions and the theorem that a function is invertible exactly when it is bijective.
Cardinality — where the course becomes surprising. Comparing sizes by bijection rather than by counting, which is the move that makes infinite sets tractable; finite sets and the pigeonhole principle; countably infinite sets, and the results students find genuinely startling — the integers, the even integers and the rationals all have the same cardinality; Cantor's diagonal argument and the uncountability of the reals, which is one of the most elegant proofs an undergraduate meets and is worth the course by itself; the Schröder-Bernstein theorem; the power set theorem and the resulting hierarchy of infinities; the continuum hypothesis mentioned as an open question resolved in an unexpected way.
Writing mathematics. The conventions — complete sentences, stated quantifiers, defined symbols, signalled proof structure, and a marked end; the standard of rigour expected and the judgement of what may be assumed; the recurring feedback that a student's proof is correct in outline and unreadable, and that this counts as incomplete because a proof is a communication; reading proofs critically and finding the gap in a plausible-looking argument; LaTeX where the course requires it, which is worth learning at some point in a mathematics degree and is as good a time as any.

Optional Topics

The axiomatic method; Zermelo-Fraenkel set theory and the axiom of choice with its equivalents and consequences.
Elementary number theory as proof practice — divisibility, primes, the Euclidean algorithm, modular arithmetic.
Combinatorics and counting arguments.
Ordinals, transfinite induction and cardinal arithmetic.
Gödel's incompleteness theorems, informally and accurately.
Introductory graph theory as a proof context.
Proof assistants — Lean, Coq or Isabelle.
The history and philosophy of the foundations of mathematics.

Resources & Tools

How to Prove It: A Structured Approach by Daniel Velleman — the best book on this material and the one most likely to change a struggling student's experience. It teaches proof construction as a structured process driven by the logical form of the statement, which is exactly what beginners lack.
Book of Proof by Richard Hammack — free and legally downloadable from the author's site, complete, well written and widely adopted. If cost is a consideration, this is the answer.
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni and Zhang — the most widely assigned commercial textbook for this course.
An Introduction to Mathematical Reasoning by Peter Eccles; The Elements of Advanced Mathematics by Steven Krantz.
Naive Set Theory by Paul Halmos — short, classic, and the reference for the set theory beyond what a transition course covers.
Free and useful:
- Hammack's Book of Proof, above — the standout.
- forall x by P.D. Magnus for the logic half, also free.
- The Open Logic Project — free, more advanced, for students who want the metatheory.
- Art of Problem Solving forums and Mathematics Stack Exchange — useful for seeing how proofs are discussed, with the caveat that reading solutions is not practising.
Tools: LaTeX for typesetting — Overleaf is free in the browser and is how mathematics is written professionally; proof assistants (Lean, with its Natural Number Game as a genuinely enjoyable free introduction) if your course goes that way.
⚠ The study method, and this one is not optional. Proof is learned by attempting proofs, failing, and being told why. Reading a completed proof feels like understanding and is not — the gap becomes obvious the first time you face a blank page. Specific advice that works: attempt every problem before looking at any solution; when stuck, write down what you are assuming and what you must show, which frequently reveals the route; work with other students, since explaining a proof aloud exposes gaps faster than anything else; and go to office hours early and often — this is the course where instructors most expect it and where a specific confusion is most quickly fixed.
Professional context: the Mathematical Association of America and the American Mathematical Society, both with student membership; the Putnam competition and MAA problem collections for students who enjoy the material.

Career Pathways

This course is a gateway rather than a destination — but it is the gateway, and its effects are large.

Mathematicians (SOC 15-2021) — research and applied positions; generally requires a graduate degree.
Statisticians and Data Scientists (SOC 15-2041, 15-2051) — the largest destination for mathematics graduates, and one where the ability to reason precisely about assumptions is what separates an analyst from a button-pusher.
Actuaries (SOC 15-2011) — a defined examination pathway, strong compensation, and a natural fit for mathematics majors.
Operations Research Analysts (SOC 15-2031) — optimisation and modelling.
Software Developers and Computer Scientists (SOC 15-1252, 15-1221) — algorithms, correctness arguments, type systems and formal verification are all proof, and this course's content is the direct foundation.
Cryptography and information security (SOC 15-1212) — cryptography is applied number theory and its security claims are theorems; the National Security Agency is among the largest employers of mathematicians in the country.
Financial analysts and quantitative finance (SOC 13-2051) — the quantitative roles require exactly this kind of rigour.
Teaching (SOC 25-2031, 25-1022) — secondary mathematics is a persistent Florida critical shortage area, with loan forgiveness implications; postsecondary requires graduate work.
Engineering and physical science roles generally; law, where mathematics majors post strong LSAT results for reasons connected to this course.
Graduate study in mathematics, statistics, computer science, economics or operations research — and performance in this course is one of the earliest and clearest signals of aptitude for it.

The Florida picture: defence, aerospace and simulation in Orlando and on the Space Coast, which employ mathematicians and operations researchers directly; financial services in Miami, Tampa and Jacksonville; insurance and actuarial work, substantial given the state's property insurance market; healthcare analytics; the university research centres; and secondary teaching, which is a documented shortage.

The advice specific to this course, and it is about a decision rather than a skill. How you find this course is genuinely diagnostic. A student who enjoys constructing proofs — who finds the struggle interesting rather than merely difficult — has good evidence for continuing in pure mathematics or theoretical computer science. A student who works hard, does adequately, and dislikes it has equally useful evidence, and should look at statistics, actuarial science, operations research, data science or mathematics education, all of which are excellent careers that make less daily use of proof. That is not a lesser outcome and the course is doing its job either way. The mistake is concluding you are bad at mathematics, when what you have discovered is which kind of mathematics suits you.

Special Information

⚠ The prerequisite is Calculus II, and the reason is not the calculus

UWF requires MAC 2312 — Calculus II — with an asterisk indicating it may be taken concurrently.

Almost none of the calculus is used. The prerequisite is not there because the course needs integration techniques; it is a proxy for mathematical maturity — evidence that a student has worked through a demanding sequence, is comfortable with abstraction and notation, and has the persistence the course requires. Some institutions require only Calculus I, and some accept any calculus with instructor permission.

⚠ The concurrency permission is worth noticing and using. Taking this course alongside Calculus II rather than after it can save a term in a tightly sequenced mathematics degree, because this course gates the upper division.

What actually helps is comfort with symbolic manipulation, a tolerance for abstraction, and — above all — a willingness to sit with a problem you cannot immediately solve. Students who are strong at computation and impatient with ambiguity find this course harder than students who are moderate at computation and patient, which is a genuinely different selection than earlier courses have applied.

⚠⚠ This course gates the upper division — take it as early as possible

MHF3202 is the prerequisite, formally or in practice, for essentially every proof-based mathematics course — real analysis, abstract algebra, topology, number theory, advanced linear algebra. It is the bottleneck of the mathematics major.

Three consequences.

Take it in the sophomore year, concurrently with Calculus II if the concurrency permission allows. A student who defers it to the junior year compresses the entire upper division into two years and may not be able to fit the sequence at all, since upper-division mathematics courses frequently run once a year and several are themselves sequential.

For transfer students this is the binding constraint. A student arriving with the calculus sequence complete but without the transition course cannot start the upper division, whatever the credit total says. Complete it before transferring if you can — and note that it is available at some Florida state colleges, Indian River State among them, which makes that genuinely possible.

And if you struggle, do not defer the rest of the major hoping it will be easier later. Get help in this course; the difficulty does not diminish and the material only becomes more necessary.

⚠ Titles vary, and one variant covers more

Source	Title
statewide	Sets and Logic
UWF	Set Theory and Mathematical Logic

Both are 3 credits and the content is the same. ⚠ The more useful check is what your institution calls this course generally, because the transition course goes under many names — "Introduction to Advanced Mathematics", "Foundations of Mathematics", "Mathematical Reasoning", "Discrete Mathematics" in some curricula, "Introduction to Proof" — and a transfer student looking for "Sets and Logic" may not recognise the equivalent on another institution's list.

⚠ Note particularly the discrete mathematics overlap. Some programmes satisfy the transition requirement with a discrete mathematics course, which covers much of the same logic and set theory alongside combinatorics and graph theory. Whether one substitutes for the other is a departmental decision and varies — ask before assuming, especially if you are a computer science student wondering whether your discrete mathematics course covers this, or a mathematics student wondering the reverse.

And note that UWF's section meets the College-Level Computation Skills Requirement, which is a general education designation — institution-specific, and worth confirming rather than assuming it transfers as such.

Course format and workload

Taught as a lecture with extensive problem work, frequently with proofs presented at the board by students. Assessment is almost entirely written proofs — problem sets and examinations — with the marking attending to correctness, completeness and clarity of exposition, which is a standard most students have not been held to before.

Expect eight to twelve hours a week outside class, and the distribution is unusual: a single problem can absorb two hours with nothing to show, and then resolve in five minutes. That is normal and is not a sign of failure.

⚠ The specific difficulties, and knowing them helps.

The blank page. Earlier courses always indicated a method; here the first task is to work out what kind of proof is called for. Velleman's structured approach — let the logical form of the statement dictate the proof's skeleton — is the single most useful technique available, and it turns "I have no idea where to start" into a procedure surprisingly often.
Negating quantified statements, which is mechanical once learned and is a persistent early stumbling block.
Knowing what may be assumed. Beginners either prove too much, re-deriving standard facts, or too little, asserting the step that needed the argument. Calibrating this takes the whole term and is learned from feedback.
Writing. A correct idea expressed unreadably is marked as incomplete, and students find this unfair until they try to read each other's work.
The emotional adjustment. Students accustomed to being good at mathematics frequently do badly on the first problem set, and some conclude they are not mathematicians. The first weeks of a transition course are difficult for nearly everyone, the skill develops in a step rather than smoothly, and instructors know this. It is the wrong point at which to conclude anything about your ability.

⚠ Logic in two departments — this course and PHI3130

Logic is taught in mathematics and in philosophy, and Florida carries both: this course, and PHI3130 Logic.

	MHF3202	PHI3130
Department	Mathematics	Philosophy
Prerequisite	Calculus II (UWF)	Typically none
Purpose	Preparation for upper-division mathematics — proof technique and set-theoretic vocabulary	Argument evaluation — logic as a general reasoning tool
Also covers	Set theory, relations, functions, cardinality, induction	Fallacies and argument analysis, or natural deduction, depending on the version

The propositional and predicate logic genuinely overlap. ⚠ But a mathematics major generally needs this version, because the set theory, relations, functions and cardinality material is the vocabulary the rest of the mathematics curriculum is written in, and the philosophy course does not supply it. Conversely, a student wanting to evaluate everyday arguments is better served by the philosophy course, which is also open without a calculus prerequisite.

Whether one substitutes for the other is a departmental decision. Ask.

Articulation and transfer

MHF3202 carries the same SCNS number across Florida public institutions and SCNS equivalency governs transfer of the credit. Note that it is available at some Florida state colleges — Indian River State among them — which makes it possible to complete the transition course before transferring, and that is worth doing given how comprehensively it gates the upper division.

Two cautions. The naming variation above means the equivalent course may not be obvious on another institution's list — check by content, not by title. And whether a discrete mathematics course satisfies the requirement varies by department; confirm with the receiving institution rather than assuming.

AI Integration

Mathematics is the domain where the difference between generating plausible text and establishing truth is sharpest, and this course is the best place in an undergraduate curriculum to see it.

Where the tools help a student. Explaining a definition or a concept — the difference between a codomain and a range, why the empty set is a subset of everything — where a patient explanation is genuinely useful. Suggesting a proof strategy when you are stuck, as a hint rather than a solution. Explaining a step in a proof you have read and not followed. Generating additional practice problems. And LaTeX help, which is mechanical.

⚠ Where they fail, and this is the instructive part.

Generated proofs are frequently wrong, and wrong in a specific way. They look exactly like proofs — correct vocabulary, plausible structure, confident tone — and contain steps that assert what needs to be proved, unjustified inferences, or an appeal to a result that does not exist. The most common failure is circularity: the "proof" assumes the theorem somewhere in the middle, which is invisible unless you are reading for exactly that.

And this is precisely the skill the course teaches. Being able to read a plausible-looking argument and find the gap is a course outcome, and running a generated proof through that scrutiny is a legitimate and unusually effective exercise. Mathematics is one of the few subjects where "sounds right" and "is right" can be separated definitively, and having a source of confident, well-formatted, sometimes-wrong arguments is — used deliberately — a teaching asset.

Quantifier handling and edge cases are unreliable. Nested quantifiers, the empty set, vacuous truth and boundary cases are exactly where generated arguments go wrong, and exactly where the mathematics matters.

Non-existent results get cited. Invented theorem names and misattributed results are common, and a proof resting on one is worthless.

What is genuinely happening in mathematics, and it is more interesting than the caution. Proof assistants — Lean, Coq, Isabelle — are formal systems that verify a proof mechanically, and they are sound by construction: a Lean-checked proof is correct, full stop. Substantial mathematics has now been formalised, including some very deep results, and the Lean mathematical library is a serious and growing collaborative project. Machine learning is being used to suggest proof steps within these systems, and the combination — a model proposing, a formal system verifying — is the active research direction.

The point that follows, and it is worth carrying. That architecture is an explicit admission that generation and verification are different operations. A language model produces probable continuations; it has no step anywhere that preserves truth, which is exactly why it can produce a confident invalid proof. A proof assistant does nothing but preserve truth. A student who has spent a term learning what makes an inference valid understands that distinction from the inside, and is better placed than almost anyone to judge what these systems can and cannot be trusted with.

Academic integrity. Read your instructor's policy. The point specific to this course: the problem sets are the only mechanism by which proof-writing develops, and it develops by failing at problems and finding out why. A student who submits generated proofs has skipped the struggle that is the entire content of the course — and, unusually, will find out quickly: the examination is a blank page, and being asked to explain your own proof at a board settles the question in under a minute.